SUMMARY

In the present study, we show that the fastest runners and swimmers are
becoming not only faster but also heavier, taller and more slender. During the
past century, the world record speeds for 100 m-freestyle and 100 m-dash have
increased with body mass (M) raised to the power 1/6, in accordance
with the constructal scaling of animal locomotion. The world records also show
that the speeds have increased in proportion with body heights (H)
raised to the power 1/2, in accordance with animal locomotion scaling. If the
athlete's body is modeled with two length scales (H, body width
L), the (M, H) data can be used to calculate the
slenderness of the body, H/L. The world records show that
the body slenderness is increasing very slowly over time.

The corresponding scaling laws for flying and running with air drag are
similar to Eqns 1,
2,
3,
4. These relations are accurate
within a dimensionless factor of order 1, as they were derived based on scale
analysis. In spite of this built-in approximation, they agree well with the
large body of experimental data available
(Bejan and Marden, 2006).

MATERIALS AND METHODS

Speed and body mass

We used this constructal framework to examine the evolution of speeds in
modern athletics: the evolution of the sport (winning speeds and body
metrics), not the evolution of the athletes. We focused on the two most
documented probes for men, the 100 m-freestyle in swimming and the 100 m-dash
in track. These are sprint probes, not endurance events. Sprint probes require
intense expenditure of work during a relatively short period of time.

In Fig. 1A and
Table 1 we see the evolution of
the world speed record (V) for male 100 m-freestyle swimming since
1912. Because of the theoretical scaling (i), we also researched the evolution
of the body masses of the record-breaking athletes
(Fig. 1B). Both V and
M have been increasing in time (t). By eliminating
t between Fig. 1A,B,
we found Fig. 1C, which shows
the evolution of V vs M.

RESULTS

There is scatter in Fig. 1C
because the span of the V and M data is short, much shorter
than the span of all biological cases of animal locomotion correlated in Bejan
and Marden (Bejan and Marden,
2006), where the M range was
10–6–103 kg. The shortness of the
contemporary timeframe has the effect of magnifying the scatter of the data.
Several additional factors also contribute to the scatter, for example,
technology (space age swimming suits, running shoes and chronometry),
competition environment (state of the art aquatic and track and field venues)
and changes in the rules of competition. In spite of these random variables,
the evolutionary direction of animal locomotion
(Eqn 1) is respected: as an
average, the faster swimmers are bigger. The best fit of the (V, M)
data of Fig. 1C according to
the theoretical proportionality between V and
M1/6 (cf. Eqn
1) is:
(5a)
where V and M are expressed in m s–1 and
kg, respectively. Eqn 5a was
obtained by power law regression, with R2=0.171. The
P-value is 0.028, and because it is less than 0.05, the correlation
shown in Eqn 5a is statistically
significant (Soong, 2004;
Vogt, 2005).

The P-value is even smaller if we correlate the
V–M data of Fig.
1C with a more general power law
V=aMb, in which the constants a and
b can be optimized. The best fit of this kind is:
(5b)
for which the P-value is 0.023 and R2=0.19. Note
the slight difference between the exponent 0.23 and the theoretical exponent
1/60.17. The scaling (Eqn 5b)
represents a steeper increase in V with M than in the
broad-range animal scaling (Eqn
1). Numerically, the two formulas
(Eqn 5b and
Eqn 1) agree in the M
range of humans, and for this reason the
V∼M1/6 scaling of
Eqn 5a is sufficient for
concluding that the animal scaling (Eqn
1) manifests itself in the evolution of swimming speeds
vs mass among record holders.

Swimming world records for 100 m freestyle, men: (A) speed (V)
vs time (t); (B) body mass (M) vs t; (C)
V vs M. The world record data for all the figures cover the period
1912–2008, and are listed in Table
1.

Running world records for 100 m dash, men: (A) speed (V)
vs time (t); (B) body mass (M) vs t; (C)
V vs M. The world record data for all the figures cover the period
1929–2008 and are listed in Table
2.

For 100 m dash (Fig. 2), the
trend, scatter and conclusion are the same. Eliminating t between
Fig. 2A,B, we arrived at
Fig. 2C. The power law
regression equation for the data in Fig.
2C is:
(6)
with R2=0.364. The P-value is 0.007, again
indicating statistical significance.

Noteworthy are the factors 0.72 and 4.85 in Eqns
5a and
6, respectively. These factors of
order 1 agree with the theoretical scaling
(Eqn 1), which after substituting
the values for g and ρ yields for swimming and running
V∼1×M1/6. Here `1' is the intercept of
the line plotted as log V vs log M. Also noteworthy is that
the factor for running (4.85) is greater than the factor for swimming (0.72).
This also agrees with the manner in which the empirical factor (not shown in
Eqn 1 but reported in the
Appendix) differentiates between the power-law correlations of animal speed
data for runners and swimmers (Bejan and
Marden, 2006).

Body height

The conclusion that body size has an effect on speed, Eqns
5a and
6, agrees fully with the
doctrine of animal scaling (Bejan,
2000; Hoppeler and Weibel,
2005; Weibel,
2000). Size can be expressed not only as M but also as
body height (H). In the scale analysis that led to Eqns
1,
2,
3,
4, the body was modeled in the
simplest possible way: with one length scale, which meant that
M∼ρLb3, where
Lb is the lone length scale. Accordingly, the theoretical
speed V of Eqn 1 should
be proportional to Lb1/2. In
Fig. 3A and
Fig. 4A we plotted the data of
Tables 1 and
2 by using the H of
the athletes as the length scale Lb. We determined the
best correlations of type V∼Lb1/2,
based on power law regression:
(7)(8)
where Lb is expressed in meters,
R2=0.248 and 0.433 for Eqns
7 and
8, respectively. The
corresponding P-values are 0.009 and 0.001, respectively; thus,
indicating statistical significance for both correlations. The H data
of Tables 1 and
2 are plotted vs t in
Fig. 3B and
Fig. 4B, respectively.

The proportionality between speeds and body length raised to the power 1/2
(Eqns 7 and
8) suggests a simpler way to
derive the speed–mass scaling rule (Eqn
1), much simpler than the analysis shown in the Appendix. During
each cycle of locomotion the body falls from a height of order
Lb. The time scale of the fall is of order
t∼(Lb/g)1/2. The
body falls forward to a distance of order Lb; therefore,
the horizontal velocity scale is
V∼Lb/t∼(gLb)1/2.
Combining this V scale with
Lb∼(M/ρ)1/3 we arrive at
Eqn 1.

Body slenderness

A body model that is more realistic than the single-scale model is a
cylinder of height H and diameter (width) L. The M
in this model is M=ρ(π/4)L2H. We
used the body density (ρ∼1000 kg m–3), the recorded
M and the recorded H to calculate the athlete's width scale
L=(4M/ρΠH)1/2. We then used the
recorded H and the calculated L to define the slenderness
(S) of the body:
(9)
The S values calculated in this manner are reported for each athlete
in Tables 1 and
2. Their significance becomes
apparent if we combine this two-scale body model with the locomotion model
proposed at the end of the preceding section.

The evolution of the slenderness of record holders over time: (A) 100 m
freestyle, men; (B) 100 m dash, men.

In swimming, the vertical length scale is L, the time scale of the
fall is (L/g)1/2 and the forward speed is
of order V∼(gL)1/2. Omitting
factors of order 1 (such as π/4), we combine the mass scale
(M∼ρL2H) with
S=H/L and obtain
L∼(M/ρ)1/3S–1/3
and:
(10)

In running, the vertical length scale is H, and the corresponding
scales are t∼(H/g)1/2,
V∼(gH)1/2 and
H∼(M/ρ)1/3S2/3. The
speed–mass relation that replaces Eqn
1 is:
(11)

The S effect differentiates between running and swimming. Dividing
Eqns 11 and
10 we anticipate
Vrun/Vswim∼S1/2,
which is a number of the same order as the ratio between Eqns
6 and
5a. The two-scale model also
suggests that from among athletes with the same mass, the ones with larger
S values are more likely to run fast. In swimming, the S
effect is the opposite but weaker: swimmers would be slightly faster if more
robust (smaller S).

Fig. 5A,B show the
S data plotted vs t, and indicate a weak progress toward
larger S values for both swimming and running. The best linear fits
for the two sets of data are:
(12)(13)
where t is the year, R2=0.31 and 0.13, and
P=0.001 and 0.07, respectively.

The same S data are plotted against M in
Fig. 6A,B. The data are too
sparse to yield statistically significant correlations; however, qualitatively
they suggest a slight increase in S vs M for running and a slight
decrease in S vs M for swimming.

DISCUSSION

The scaling trends revealed by the speed data suggest that speed records
will continue to be dominated by heavier and taller athletes. This trend is
due to the scaling rules of animal locomotion, not to the contemporary
increase in the average body size of humans. The mean height of humans has
increased by roughly 5 cm from 1900 to 2002
(Plastic Soldier Review,
2002). During the same century, the mean height of champion
swimmers and runners has increased by 11.4 cm and 16.2 cm, respectively
(Fig. 3C,
Fig. 4C).

The insight gained in this paper allows us to speculate what the running
speeds might have been in ancient Greece and the Roman Empire. There is no
record of what the winning speeds were then, because the competition was for
winning the race, not for breaking a time record. Chronometry did not exist.
In antiquity body masses were roughly 70% of what they are today
(Plastic Soldier Review, 2002;
Hpathy 2009;
National Health and Nutrition Examination
Survey, 1999). According to Eqn
6, this means that speeds were lower by a factor of roughly
(0.7)1/6=0.94. In other words, if the 100 m dash in military
training today is won in 13 s, 2000 years ago it would have been won in∼
14 s.

This insight also teaches us why certain training techniques are successful
in high-performance sports. For example, in modern speed swimming, the
doctrine holds that the swimmer must raise his body to the highest level
possible above the water. Two explanations are given for this swimming
doctrine: air drag is much smaller than water friction, and the water wave
generated by the body propels the body better
(Collela, 2009). The doctrine
is correct but for a different reason, which is evident in
Eqn 7. When the body is high
above the water it falls faster (and forward) when it reaches the water line.
For the same reason, the speeds of all water waves exhibit the same scale as
in Eqn 7, in which
Lb is the length scale of the wave
(Prandtl, 1969). The crest of
the wave falls with a speed of order
(gLb)1/2, which becomes visible
as the forward speed of the traveling wave.

CONCLUSION

In the future, the fastest athletes can be expected to be heavier and
taller. If the winners' podium is to include athletes of all sizes, then speed
competitions might have to be divided into weight categories. This is not at
all unrealistic in view of the body force scaling
(Eqn 3), which was recognized
from the beginning in the structuring of modern athletics. Larger athletes
lift, push and punch harder than smaller athletes, and this led to the
establishment of weight classes for weight lifting, wrestling and boxing.
Larger athletes also run and swim faster.

APPENDIX

Here is a brief summary of the scale analysis of animal locomotion, which
leads to Eqns 1,
2,
3,
4. It was first done for flying
(Bejan, 2000) and then
generalized to all locomotion: running, flying and swimming
(Bejan and Marden, 2006).

The animal body has a single length scale (Lb). Its
mass scale is MρLb3. Locomotion is
a rhythm – a sequence of cycles. During each cycle the body must perform
work in two ways, in the vertical direction (W1) and in
the horizontal direction (W2). Both W1
and W2 are destroyed. The vertical work is necessary in
order to lift the body to a height of order Lb:
(A1)

The horizontal work is necessary in order to push through and penetrate the
surrounding medium. If the body length and speed scales (i.e. the Reynolds
number) are large enough, then the horizontal work is of order:
(A2)(A3)
where Fdrag is the drag force, ρm is the
density of the medium, CD∼1 is the drag coefficient
and Lx is the distance traveled during the cycle.
Together, these scales allow us to estimate the total work per distance
travelled:
(A4)

The time scale of the cycle is the Galilean time of free fall from the
height Lb:
(A5)

The horizontal travel during the cycle is
Lx∼Vt, and
Eqn A4 becomes:
(A6)

The right side is a sum of two terms combined as
A/V+BV2, where A and B are two constants and
V may vary. This sum is minimal when
V∼(A/B)1/3, which yields:
(A7)

The frequency associated with this cycle is
t–1∼(g/Lb)1/2
or:
(A8)

The necessary body force scale F is dictated by the lifting work
W1∼FLb∼MgLb
therefore:
(A9)

The minimum work per distance traveled is obtained by substituting
Eqn A7 and
Lb∼(M/ρ)1/3 into
Eqn A6:
(A10)

In conclusion, the scaling relations (Eqns
1,
2,
3,
4) have been derived here in Eqns
A7,
A8,
A9,
A10. The modifying factor
(ρm/ρ)1/3 depends on the medium. In flying, theρ
m (air) is roughly equal to ρ/103, and the
factor (ρm/ρ)1/3 is close to 1/10. In swimming,
the ρm (water) is the same as the body density, and the factor
(ρm/ρ)1/3 is 1. In running, the modifying factor
is between 1/10 and 1, and depends on the nature of the running surface and
the speed. For example, running through snow, mud and sand is represented by a
(ρm/ρ)1/3 value close to 1. Running at high
speed on a dry surface is represented more closely by a
(ρm/ρ)1/3 factor similar to the one that
represents flying.

In summary, the effect of the factor
(ρm/ρ)1/3 is weak and of order 1, and for this
reason it was left out of Eqns 1
and 4. Important to note is that
(ρm/ρ)1/3 differentiates between locomotion
media in an unmistakable direction: if M is fixed, speeds increase in
the direction sea → land → air (cf.
Eqn A7); the work requirement
decreases in the same direction (cf. Eqn
A10). The animal speeds collected over the
M=10–3–103 kg range in
fig. 2 of Bejan and Marden
(Bejan and Marden, 2006)
confirm the differentiating effect of the surrounding medium. Each cloud of
data is approximated by:
(A11)(A12)(A13)
It can be verified that Eqn A11,
A12,
A13 agree with
Eqn 1 within a factor of order 1.
They also agree with Eqns 5 and
6.

The data collected in fig. 2
of Bejan and Marden (Bejan and Marden,
2006) also confirm that the medium factor
(ρm/ρ)1/3 does not have an effect on the
frequencies (t–1) and forces (F) of flyers,
runners and swimmers. This is in accordance with Eqns
A8 and
A9, in which
(ρm/ρ)1/3 is not present.

LIST OF ABBREVIATIONS

CD

drag coefficient

F

force, N

Fdrag

drag force, N

g

gravitational acceleration, m s–2

H

body height, m

L

body width, m

Lb

single body length scale, m

Lx

distance traveled, m

M

body mass, kg

P

probability of true null hypothesis

R2

coefficient of determination

S

body slenderness, H/L

t

period, s

t

time, years

t–1

frequency, s–1

V

speed, m s–1

W

ork, useful energy, J

W1

work done vertically, J

W2

work done horizontally, J

ρ

body density, kg m–3

ρm

density of the medium, kg m–3

FOOTNOTES

J.C. is the starting 100 m breaststroke swimmer on Duke University's NCAA
swimming team. A.B.'s research on the constructal law of design in nature is
supported by grants from the US Air Force Office of Scientific
Research and the National Science
Foundation.

Greenewalt, C. H. (1975). The flight of birds:
the significant dimensions, their departure from the requirements of
geometrical similarity, and the effect on flight aerodynamics of that
departure. Trans. Am. Philos. Soc.65, 1-67.

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